ballistic missiles notes

8/12/2019 Ballistic Missiles Notes

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AT 6

BALLISTI MISSIL TRAJETORIES

Tis a priiple o war at we yo a se telitig tis better tha ao

Napoleo

61 HISOIAL AKODWile te prist igt isist tat e istory o ballisti issiles

stretes bak to te rst se o rde rokets i warare by eCiese, lograge ballisti issies, wi oer s i is apter,ae a ery sort istory

Te ipets or deelopig te lograge roket as a weapo o warwas iroally, te Treaty o Versailles wi eded World War orbade te Geras o deelop lograge arillery As a reslt eGera ig Coad as ore reeptie to sggestios or roketdeeopet a were ilitary oads i oter ories Te

reslt is well kow orts , beg i 93 der te diretio o teCaptai Walter orberger liated i e rst sessl la oa A4 ballisti issile (ooly kow as te V) o 3 ober94 rig 943 ad 944 oer 80 test issiles ere ired roeeemde Te rst two operatioa issiles were ired agaist ariso 6 Sepeber 944 ad a attak o odo ollowed days laterBy te ed o e war i May 945 oer 3 ,000 V s d bee red ager

Geeral orberger sed p te se o te lograge issile iWorld Wa as too ate

e igt ae added tat it was o apartilarly eetie weapo as sed t ad a dispersio a e targeto 0 iles oer a rage o 00 iles ad arried a waread o abot

7


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278 B L L S T SS L T CT O I S Ch

oe o o e g explosve Aato3 Neverteless, t was oe ta

s a exteso o lograge artlleryt was te st ballst issile

as we kow t todyAer te war tere was a ad srable to aptre" te Gerasietsts o eeede wo were resposble or tis teologalrale Te Uited States Ary obtaied te serves o ost o tekey setists ad teias ildg orberger ad Werer voBra Te Sovets were able to asseble at Kiki a sta o abot 80e der te orer proplso expert Werer Ba Tey wereassiged te task o desigig a roket otor wit a trst o 260 ,000ods ad later oe o 530 ,000 pods trst 3

At a telligee breig at WrigtPatterso Air ore Base iAgst 52, ese dsqietg ats were revealed by orberger woad iterviewed ay o is orer olleages o a retr trp toGeray ve ore disqetg sold ave bee te report tat teSoviets ad bilt a separate atory dig adaet to tat opiedby te Gera workers No Gera was peritted to eter tisseparate bildig

Te experts displayed o partlar sese o ieday Ater all,

te ato bob was stll too eavy to be arred by a roketAordg to r arol roa o te os Alaos Sietiaboratory te key qesto i te 950s was We old te ACoe p wit a waread ligt eog to ake issiles pratial?"

Te ballist issle progra i tis otr \S still essetially iabeyae we a liited otrat or te CBM alled Atlas" wasawarded Bt i Noveber 952 at wetok te teolearMke" sot eded all dobts ad paved te way or te Srip" sot

o Mar 954 w revoltoied te prograAordigly , i Je 953 , revor Garder , Assistat Seretary o

te Air ore or Resear ad eveopet , oveed a speal gropo te atos leadig sietsts kow as te Teapot Coittee egrop was led by te late roessor Jo vo Nea o te stitteor Advaed Stdies at reto ey et a vaat r iglewood, Calora, ad te reslt o teir stdy was areoedato to te Ar ore tat te balst ssile progra be

reatvated wt top priort tober 95 3 a stdy otrat was plaed wit teRaoWooldridge Corporato ad by May 954 te ew CBMprogra ad igest Air ore prority y 95 4, Brigadier Geeral


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Se 62 E E L B I S I M I SS I E O B 27

Berard A Srieer was gi te mometa ask o direti taeerated CBM program ad bega adpikig a sta o miitary

assistats We e reported to te West Coast e ad wit im aes o or oersaog tem ieteat Cooe Beami PBasigame, ater to beome te irst ead o te eartet oAstroatis at te Uited States Air ore Aademy

Witi a year o its begii i a oerted paroia soobidig i gewood te progra d passed ro top Air orepriority to top atioa priority rom 2 mai otrators at teegiig, te progra ad, by mid959, 30 mai otrators admore ta 80 ,000 peope patiipatig direty

Progress was rapid Ater tree sess attempts, te irstsess igt o a Series A Atas took pae o 7 eember 957y 4 mots ater te Soiet Uio ad aoe tat i t ad aiterotieta baisti missie , Atas was a reaity

We te istory o te baisti missie is bot iterestig adsigiat, a kowedge o ow ad wy it works is idispesabe orderstadig its epoymet as a weapo

e traetory o a issie diers rom a steite orbit i oy oe

respetit itersets te srae o te rt terwise, it oows aoi orbit drig te ree igt portio o its traetory ad we aaye it s beaior aordig to priipes whi yo aready kow

Baisti issie targetig is st a speia appiatio o te Gasspobem wi we treated rigorosy i Capter 5 tis apter wewi preset a somewat simpied saar aaysis o te probem sotat yo may gai some res isigt ito te atre o baistitraetories o ompte preise missie traetories reqires te

ompexity o pertrbatio eory I this apter we are oeredmai wit oept s

62 HE GENEL BALLIST MSSL ROBLMA baisti missie traetory is omposed o tree partste powered

igt porto wi asts ro a to trst to or brot, tefreight porto wi ostittes most o te traetory, ad tereentry portio wi bgis at some i deed poit were

atsphei drag eomes a sigiat oe i deteriig theissie 's pat ad asts ti impatSie eergy is otiosy beig added to te missie drig

powered igt we aot se 2 body meais to determie its pat


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20 B I C J C O Ch

rom a o ro e pa o e missie drig is riia paro e ig i s deermied by e gidae ad aigaio sysem is

s e opi o a eire orse ad wi o be oered ererig reeig e raeory is par o a oi orbiamos

aways a eipsewi we a aaye sig e priipes eared iCaper .

Reery ioes e dissipaio o eergy by riio wi eamospere . wi o be disssed i is ex

We wi begi by assmig a e ar does o roae ad ae aide a wi reery sars is e same as e bro aide

is aer assmpio isres a e eeght ajectory ymmecal ad wil aow s o derie a airy simpe expessio ore reei rage o a issie i erms o is bro odiios

We wi e aswer a more praia qesiogie r' vbo ada desired reeig rage, wa ipa age a bro isreqired?"

oowig a disssio o maximm rage raeories, we wildeermie e imeoi or e reei porio o e raeory

621 eometry of te Trajector. Sie yo are aready amiiarwi e ermioogy o orbia meais, s erms as eig abro," eig o apogee," ipa age a bro," e,eed o be redeie ere are , oweer, a ew ew ad amiiaerms wi yo ms ear beore we embark o ay deriaiosigre 62. dees ese ew qaiies

62.2 e ondimensiona ameter, We wi id i eryoeie o deie a odimesioa paramee aled s a

6 2 )

a b e eaaed a ay poi i a orbi ad may be og o ase sqared raio o e speed o e saeie o irar saelie speeda a poi. ie V =

(622)


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Sec. 62 G N E R A L BA L L ST I C M SS I L E P R O B L 281

- powered ight range angle

' - free-ight range angle

n - re-entry range angleA - total range angle

poit

R - ground range of powered

itRf - ground range of free

ight

Rre ground r ange of reentryRt - total ground range

Figre 6.2-1 Geometry of e ballistic missile trajectory


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282 B L I ST C SS E T CTO ES Ch

Te e o o ott or tete bt re ro pot topot e orbt We Q eql to te tellte extl loalrlr telte peed T odto Q 1 ) ext t eery pot rlr orbt d t te ed o te or x o every ellptl orbt

Q 2 t e tt te tellte extly epe peed d o prbol orbt Q greter t , te telte o a yperolorbt t wold be rare to d bllt le wt Q eqal to orgreter th

We tke te lr eergy eqto o Cpter ,

d bttte or the expreo pOr ro eqto (6 - ) Thyeld bot o the ollowg reltop w will proe el:

(6-3)

or I Q = 2 - 1 (6-4)6.23 FreeFight Range quation Se te ree-igt

tretory o ile o eto te geeral eqto o o

be ppled to te brot pot

r =p .

bo 1 + e cos vbo

Solg or o vbo we get

p - rbcosvbo=_0

erbo

(62-5)

(6-6)


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G E E L B L I ST I C I SSI E O B E 8

ire 622 Symmetrica traj ectory

Since the reeight trajectory is assumed to e symmetrica h =hre ha the reeight range ange ies on each side o the majoraxis , and

co = cos vo

.

Eqation66) aove can, thereore, e written as

(67


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4 BI S C SS S

,T r pos = _ ro

(6.28)

We now have an expression for the free-ight range angle i terms ofp, e and ro Since p = / and =c , we can use the denitionof to obtain

- r2 v2 co2 ,'-= = rQ C

Now, since p = 1 - e

e = 1 Q Substiuting p = rQ c and =n' we get

e = 1 + ( ) os

(62-9

(6_2-10)

(6 2- 1 1 )

If we now substitute equations (6 2-9) and (6 .2- 1 1 ) into equation(6 2-8 we have one form of the ee-ight ange equation:

(6212

From this equation we can calculate the free-ight range angle resultingfrm any given combination of burnout condiions, ro' o and "

While this will prove o be a very valuable equation, it is notparticularly useful in solving he typical ballistic missile problem whichcan be stated thus : Given a particular launch point and target , the total

range angle, can be calculated as we shall see later in this chapte Ifwe know ho far the ssile will ravel during powered it andreentry , the required free-ight range angle, ', also be comes known . Ifwe now specify ro and o for the ssile , what shold the it-pathangle, be in orde that the missile will hit the target ?


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ec G N A A L I L P L 5

oer words, wold be e ae a eqao or ers o ad

We old, w a lle algebra, oose e sragorward way osolg e rage eqao or os sead, we will dere aexpresso or geoerally bease detae soe raereresg geoeral properies o e ellpse

\

\

ige 623 Ellipse geoery

62 Te ltat Agle Eatio gre 3 we aedraw e loal oroal a e bro poi ad also e agead oral a e bro po Te le ro e bro poi o seodary os, F s alled rb .

Te agle bewee e loal orioal ad e age (direio o s e gpa agle,

Se r

is perpedlar o e loal

oroal, ad e oral s perpedlar o e age, e aglebewee ad e oral s also

Now i a be proe (altog we wo do i a e agle


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2 A A S

betwee O ad o is biseted by te oma is at gies ise tomay iteestig appiatios o te eipse t meas, o exampe,

tat, i te eips epeseted te sae o a mio, it emaatigo oe os wod be eeted to te ot o sie te age oeetio eqas te age o iidee te eiig o a oom weemade i te sape o a eipsid, a pe so stadig at a patia poiti te oo oespodi to oe os could be heard cerly by apeso stadig at te ote os ee tog e wee wispeigis is, i at , te basis o te soaed wispeig gaey"

Wat a tis meas o o deiatio is simpy tat te agebetwee

ad

ois

2bo'et s oetate o te tiage omed by F F ad te botpoit We kow two o te ages i tis tiage ad te tid a bedeteied om te at tat te ages o a tiage s to 1 80 we diide te tiage ito two igt tiages by te dased ie, dsow i ige 6 4, we a expess d as

d = rb I o 2ad aso as

6 1 3)

6 1 4)

Combiig tese two eqatio ad otig tat si ( 1 80 - x = x, we get

(2bO + rO 2 rbo 2

6 1 5)

Sie bo a2 0 om eqtio 6 3) ad + bo 2

s i n

+ ) =2 - Qbo s i n .bo 2 Q 2bo (66)

s is aled te ightpath angle euation ad it poits ot someiteestig ad impotat ats abot b aisti missie taetoie s


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Sec 62 G N A A L 7

gure 62 ipse geoety

Sppose we wat a ssl to trael a ree-igt rage 90 ad itas Qo= 9. Sbstttg tese ales ito eqatio 6 2 6) ges s

si n( 2


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2 A L L I I I I L A I

Te fa a ere are wo raeores o e arge sold osrprise o se ee ery sorrage balls raeores eb s

proer A amliar llsrao of s resl s e beaor of waerdsarged from a garde ose osa waer ressre ad ozzlesg e speed of e waer leag e ozzle s ed If a argewell w e mamm rage of e ose s seleed e arge abe by a la or loed rae ory

Te are of e g ad low raeory depeds primarly o eale of Q I

Q s less ere wll be a lm o ow large may be order a e ale of e rg sde of eqao (6 6)does o eeed Ts mples a tere a axmum range for ame wt Qo e tan Ts mamm rage wll always be lessa 80 for bo less a roded a s aaable ere wllbe bot a g and a ow aectory to te target

If Q s ealy oe of e rae ore s o e arge wll be erlar orb oeg e bro ad reery pos Ts woldo be a ery raal mssle raeory b does represe eborderle ase were rages of 80 ad more are s aaable

If Q s greaer a eqao (6 - 6) wll always yeld oe

pose ad oe egae ale for regardless of rage A egae s o praal se e raeory wold eerae e ear soony te ajectory can be reazed for Q eater tn

Te real sgae of Qgreaer a s a rages eess of 80 are possble A llsrao of s a raeory wold be amissle dreed a e Nor Amera oe from Asa a eso ole le s a raeory wold aod deeo by ororer radar fees wold be osly erms of payoad

delered ad aray aaable Neereless e shok ale of sa srprse aak i erms of wa mg o owards reaig aosamog or defese fores sold o be oerlooked by mlaryplaers

Sie bo e g ad low raeores resl from he same ad ey bo ae e same eergy Bease a = - P2& e maoras of e gh ad low ra eores are e same leg

Table 6 2 - 1 shows whch trajectories are possble for varous

ombaos ofQ

and '. F

gre 6 .2-5 sold be elpfl nvsualizngeachcase .


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Sec. 62 G N R A AL I ST C M SS I E P R OB L E M 289

Imposs ib le

Fgr 6.2-5 Example ballistic missile rajectoies


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29

0 < 10 = > 1

A A

Siic of o

' < 1 8 0

both hgh and low f'


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Se L B L L I ST I C S I L O B L E

sg eqao (6 os + N

R ff ( 36 4Deg ) ( 6n m i ) , 1 84n miDeg

9

EAME RBEM A mssiles oordaes a bro are3 0N 60E Reery s laed for 30S 60 Bro eloy ad

ae ae DUU ad 05 D reseely ' s less ha 80 ha ms he ghah agle be a bro?

Beore we a se he iah agle eqao o d


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292

e

8.Q-

c.-E

;0

: 20-

I

ballistic missiles notes

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